Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.
Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.
The quantum theory of the third virial coefficient C is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of CBE are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for He4 and He3, by means of a comparison of the corresponding expansion for the second virial coefficient B with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to B. Then the leading term of C is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for C is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.
We investigate several approximations for self-bound many-fermion systems, each of which employs the Green's-function formalism together with a suitable factorization to close the otherwise infinite hierarchy of coupled integral equations. The particular approximations, which attempt to include the important two-body correlations for short-range forces, differ in the choice of propagator $\ensuremath{\Lambda}$ for two-particle intermediate states in the T matrix used to calculate the two-body correlation function. The advantages and shortcomings of the choice ${\ensuremath{\Lambda}}_{00}\ensuremath{\sim}{{G}_{1}}^{0}{{G}_{1}}^{0}$ have been noted previously and are not treated here. We show that the choice ${\ensuremath{\Lambda}}_{10}\ensuremath{\sim}{{G}_{1}}^{0}{G}_{1}$, which appears in a "natural" factorization, is formulated in a manner which preserves much of the simplicity of the ${\ensuremath{\Lambda}}_{00}$ approximation. (Numerical calculations using the ${\ensuremath{\Lambda}}_{10}$ approximation are presented by Foster in the following paper.) The ground-state energies for three $\ensuremath{\Lambda}$ approximations (${\ensuremath{\Lambda}}_{00}$, ${\ensuremath{\Lambda}}_{10}$, and ${\ensuremath{\Lambda}}_{11}\ensuremath{\sim}{G}_{1}{G}_{1}$) are expanded through third order in the interaction strength to facilitate comparison with each other and with the Brueckner-Goldstone series. Numerical comparison of these third-order contributions for a square-well potential is presented. ${\ensuremath{\Lambda}}_{10}$ is in dramatically closer agreement with the full perturbation theory (for ${{k}_{F}}^{0}c\ensuremath{\lesssim}1$) than are the other $\ensuremath{\Lambda}$ approximations or Brueckner theory.
In the framework of real-time Green's functions, the general kinetic equations are investigated in a first-order gradient expansion. Within this approximation, the problem of the reconstruction of the two-time correlation functions from the one-time Wigner function was solved. For the Wigner function, a cluster expansion is found in terms of a quasiparticle distribution function. In equilibrium, this expansion leads to the well-known generalized Beth-Uhlenbeck expression of the second virial coefficient. As a special case, the T-matrix approximation for the self-energy is investigated. The quantum kinetic equation derived thus has, besides the (Markovian) Boltzmann collision integral, additional terms due to the retardation expansion which reflect memory effects. Special interest is paid to the case that bound states exist in the system. It is shown that the bound state contribution, which can be introduced via a bilinear expansion of the two-particle T matrix, follows from the first-order retardation term in the general kinetic equation. The full Wigner function is now a sum of one function describing the unbound particles and another one for the bound state contribution. The latter two functions have to be determined from a coupled set of kinetic equations. In contrast to the quantum Boltzmann equation, energy and density of a nonideal system are conserved. \textcopyright{} 1996 The American Physical Society.
All the three-electron wave functions and energy levels are obtained for the one-dimensional Hubbard model. The second and third virial coefficients for the system are calculated exactly. The virial expansion is converted into an electron density expansion which appears to converge much better than the virial series. The two-particle and three-particle contributions to the specific heat, entropy, intra-atomic correlation function, and magnetic susceptibility of the system are obtained and presented graphically.
Within the conserving Φ-derivable and virial approaches to equilibrium dilute strongly-interacting systems we study a system of non-relativistic fermions of one kind (e.g., neutron matter) interacting via a pair potential in the limit nλ3≪1 , where n is the fermion number density and λ is the thermal wavelength. The variety of potentials of the quantum and classical origin is used for the quantitative comparison of results obtained within both approaches. In all cases relevant to the nucleon-nucleon interaction, the attraction volume of the interaction is larger than the repulsion one. General properties of the Φ derivable approach are studied. Expressions for the second and third virial coefficients are derived and analyzed for the Φ functional including the tadpole and sandwich diagrams. The classical high-temperature limit of the Φ functional is reproduced by the tadpole diagram. Next, classical, semiclassical (up to order ℏ3 ), and purely quantum (Beth-Uhlenbeck) virial expansions of the equation of state are studied. Various extrapolations of the virial equation of state are considered including the van der Waals form and the excluded volume models. We derive the expression for the second virial coefficient within the effective range approximation for the scattering amplitude. Comparison with purely quantum Beth–Uhlenbeck result employing experimental phase shifts shows approximate agreement for low temperatures, T≲20 MeV. Then, we study the problem of the anomalously large empirical value of the nucleon-nucleon scattering length, appearing due to existence of the quasi-bound state in nucleon-nucleon scattering. The latter is destroyed already in a dilute matter due to the Pauli blocking. Subtracting the quasi-bound state term, we show that the second virial coefficient has the same low temperature dependence as one obtained within the Φ derivable approach in the Born approximation. Also, we discuss origins of differences arising beyond the region of the common validity of the Φ derivable and virial approaches.